Physics Friday 82

One might recall my previous post where I considered a system in contact with a thermal reservoir with which it could exchange energy; from considering the entropy of the system+reservoir combination as a function of the system’s energy to derive the Boltzmann factor. This method of analysing the statistical system, by considering the different probabilities of different states of a system when it is in contact with a thermal reservoir with which it can exchange energy is known as statistical mechanics in “canonical formalism” or in “Helmholtz representation.”
In particular, we noted that for system energy E1 and total energy E, we could expand the entropy of the system in Taylor series about E and solve for the number of reservoir microstates N2:
or in terms of the thermodynamic beta ,
where S2(E) is the reservoir entropy when the total energy is entirely in the reservoir. The latter exponential factor, the only one which is a function of our system’s energy, is the Boltzmann factor.
We noted that the probability of the system having energy E1 is proportional to N2. In particular, if we let N be the total number of microstates of the system+reservoir at total energy E, then the probability that the system is in a state of energy E1 is
and from the Boltzmann definition of entropy, the total entropy is,
so
Let U be the average value of the energy of our system. Then S(E)=S1(U)+S2(E–U), and expanding our reservoir entropy in Taylor series about E–U (our equilibrium point) instead of E,
so that
And so
and due to the additivity of the entropy of the system and reservoir:,
so the probability becomes,
where is the Helmholtz free energy of our system, and is, again, the Boltzmann factor.
We do not yet know the Helmholtz free energy, but we can compute it from the above. We note that the exponential it appears in plays the role of a normalization factor: Summing the probability over all allowed states of our system, where in state s the system has energy Es, we get:
so
we denote this sum, called the “canonical partition function,” by Z. If we consider the sum over energy levels Ei, then we have,
where gi is the degeneracy of the energy level Ei.
[In classical mechanics, the parameters are of a particle are continuous, and we can’t actually express the partition function as a sum, having to replace it with an integral with a “coarse graining” procedure (see here). In quantum mechanics, however, energy levels are discrete, and the above summation makes sense.]

Now, there are a number of thermodynamic variables we can extract from Z. First, as , we see .
Next, consider the average energy U. This is the expectation value of the energy, and will be the internal energy of our system (thus the choice to name it U):
Now, note that, so that,
or, using the chain rule to rewrite β in terms of T, we have.
Consider the second derivative of with respect to β:,
the variance of the energy.
Now, the heat capacity at constant volume is.
Using the latter expression, we can find the dimensionless specific heat capacity (at constant volume):.

[…] of the reservoir and total entropies (as functions of energy and particle number): . As with our canonical formalism, we can expand the entropy as a Taylor series in energy about Et-U and in particle number about the […]